CBSE Class 10 Mathematics 2026: Complete Chapter-wise Formula Sheet, Important Theorems, and Scoring Tips for Board Exams

Class 10 Mathematics is simultaneously one of the most feared and most scoring subjects in CBSE board exams. Students who approach it with the right formula sheet, clear understanding of theorems, and practiced problem-solving patterns consistently score 90+. This guide provides a comprehensive chapter-wise formula and theorem reference for Class 10 Maths, with exam-specific scoring tips that most students miss.

How Class 10 Maths Paper is Structured (CBSE 2026)

The Class 10 Mathematics paper is 80 marks (theory) + 20 marks (internal). Theory paper: Section A has 20 objective questions of 1 mark each; Section B has 5 short answer questions of 2 marks each; Section C has 6 short answer questions of 3 marks each; Section D has 4 long answer questions of 5 marks each; Section E has 3 case study questions of 4 marks each.

Total: 80 marks. Passing requires 26/80 in theory. Scoring 70+ (87%+) requires strong performance across all sections. Scoring 80/80 is achievable with rigorous preparation — several students score full marks in Mathematics every year.

Chapter 1: Real Numbers

Key Theorem: Fundamental Theorem of Arithmetic — every composite number can be expressed as a product of primes in exactly one way (ignoring order). Example: 36 = 2² × 3².

HCF and LCM: HCF × LCM = Product of two numbers (valid only for two numbers, not three). HCF using Euclid’s Division Lemma: a = bq + r, then HCF(a,b) = HCF(b,r). LCM = (a × b) / HCF(a,b).

Irrational numbers: Proof that √2, √3, √5 are irrational using contradiction (assume rational, then show denominator divides numerator, leading to contradiction). Examiners love 3-mark proofs of irrationality.

Terminating decimals: p/q terminates if q = 2^m × 5^n (no other prime factors).

Chapter 2: Polynomials

Zeroes of polynomial: For a quadratic ax² + bx + c with zeroes α and β: Sum of zeroes (α + β) = -b/a. Product of zeroes (αβ) = c/a. Quadratic with given zeroes: x² – (α+β)x + αβ = 0.

For cubic ax³ + bx² + cx + d with zeroes α, β, γ: α + β + γ = -b/a; αβ + βγ + αγ = c/a; αβγ = -d/a.

Division algorithm: Dividend = Divisor × Quotient + Remainder. If remainder is zero, divisor is a factor.

Chapter 3: Pair of Linear Equations in Two Variables

Consistency conditions for a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:

• Unique solution (consistent): a₁/a₂ ≠ b₁/b₂
• Infinitely many solutions (consistent, dependent): a₁/a₂ = b₁/b₂ = c₁/c₂
• No solution (inconsistent): a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Methods of solution: Substitution, elimination, cross-multiplication. Cross-multiplication formula: x/(b₁c₂ – b₂c₁) = y/(c₁a₂ – c₂a₁) = 1/(a₁b₂ – a₂b₁).

Word problems tip: Set up variables clearly before writing equations. Define “let x = …” and “let y = …” explicitly. Examiners award method marks even if the final numerical answer is wrong.

Chapter 4: Quadratic Equations

Standard form: ax² + bx + c = 0, a ≠ 0.

Quadratic formula: x = [-b ± √(b² – 4ac)] / 2a.

Discriminant (D = b² – 4ac):

• D > 0: Two distinct real roots
• D = 0: Two equal real roots (x = -b/2a)
• D < 0: No real roots

Factorisation method: Split the middle term (find two numbers whose product = ac and sum = b). Practice 20+ factorisation problems to develop speed.

Chapter 5: Arithmetic Progressions

General term: aₙ = a + (n-1)d, where a = first term, d = common difference.

Sum of n terms: Sₙ = n/2 [2a + (n-1)d] = n/2 [a + aₙ].

Common tricks: If three terms are in AP, assume them as a-d, a, a+d (sum gives 3a). If four terms are in AP, assume them as a-3d, a-d, a+d, a+3d. This trick eliminates one variable and simplifies equations significantly.

Chapters 6 & 10: Triangles and Circles

Basic Proportionality Theorem (Thales’ Theorem): If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. Converse is also a theorem — if a line divides two sides proportionally, it is parallel to the third side.

Criteria for similarity: AA (Angle-Angle), SSS (Side-Side-Side), SAS (Side-Angle-Side). In similar triangles: ratio of areas = (ratio of corresponding sides)². This relationship is tested almost every year.

Pythagoras Theorem: In a right triangle, the square of the hypotenuse = sum of squares of the other two sides. Converse: if a² + b² = c², the angle opposite c is a right angle.

Circles: Tangent to a circle is perpendicular to the radius at the point of contact. Lengths of tangents drawn from an external point to a circle are equal. Angle in a semicircle = 90°. Angle at centre = 2 × angle at circumference on the same arc.

Chapters 8 & 9: Introduction to Trigonometry and Applications

Basic ratios: sin θ = P/H, cos θ = B/H, tan θ = P/B, where P = perpendicular, B = base, H = hypotenuse.

Reciprocal relations: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.

Pythagorean identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Standard angle values (memorise completely):

Height and Distance (Ch. 9): Always draw a diagram first. Angle of elevation = angle measured upward from horizontal. Angle of depression = angle measured downward from horizontal. Two classic problem types: (1) finding height of a building given angle of elevation and distance; (2) finding distance between two objects viewed from a height. Practice at least 15 problems of each type.

Chapter 11: Areas Related to Circles

Area of circle = πr². Circumference = 2πr. Use π = 22/7 unless told otherwise.

Area of sector (with angle θ) = (θ/360) × πr². Length of arc = (θ/360) × 2πr.

Area of segment = Area of sector – Area of triangle.

Common combinations: Area of ring (annulus) = π(R² – r²). Area of shaded regions in composite figures requires careful subtraction — identify which areas to add and which to subtract.

Chapter 12: Surface Areas and Volumes

Formulas for all shapes must be memorised completely:

• Cylinder: CSA = 2πrh, TSA = 2πr(r+h), Volume = πr²h
• Cone: CSA = πrl (l = slant height), TSA = πr(r+l), Volume = (1/3)πr²h
• Sphere: SA = 4πr², Volume = (4/3)πr³
• Hemisphere: CSA = 2πr², TSA = 3πr², Volume = (2/3)πr³

The most common exam question type: a shape is converted into another (e.g., a metallic sphere is melted and recast into cones). Set up: Volume₁ = n × Volume₂ and solve for n or the unknown dimension.

Chapter 14: Statistics

Mean: Direct method: Mean = Σfx / Σf. Assumed mean method: Mean = a + Σfd/Σf (where d = x – a). Step deviation method: Mean = a + h × Σfu/Σf (where u = (x-a)/h).

Median = l + [(n/2 – cf)/f] × h. Where l = lower class limit, n = total frequency, cf = cumulative frequency before median class, f = frequency of median class, h = class width.

Mode = l + [(f₁ – f₀) / (2f₁ – f₀ – f₂)] × h. Where f₁ = frequency of modal class, f₀ = frequency of class before modal class, f₂ = frequency of class after modal class.

Draw ogives (cumulative frequency curves) for finding median graphically — a standard exam question worth 4-5 marks.

Chapter 15: Probability

Probability = Number of favourable outcomes / Total number of outcomes.

P(A) + P(not A) = 1. Probability always lies between 0 and 1 inclusive.

Standard problems: drawing cards from a deck (52 cards: 4 suits × 13 cards each), throwing dice (6 outcomes per die), selecting coloured balls from a bag. Know how many cards of each type are in a standard deck (4 aces, 12 face cards, 26 red, 26 black).

Final Exam Strategy

In the exam: read all questions in the first 10 minutes. Start with the questions you are most confident about to secure those marks first. Show all working for every question — partial marks are awarded for correct method even if the final answer is wrong. In case study questions, attempt all sub-parts even if unsure — the questions are interconnected and even one correct sub-part earns marks. Leave no question blank.

Time allocation guide: Section A (20 marks) — 30 minutes. Section B (10 marks) — 20 minutes. Section C (18 marks) — 35 minutes. Section D (20 marks) — 40 minutes. Section E (12 marks) — 25 minutes. Review — 10 minutes. Total: 160 minutes for an 180-minute paper.